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          测量电源电动势E和内阻r
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        <h3 id="知识储备"><a href="#知识储备" class="headerlink" title="知识储备"></a>知识储备</h3><ul>
<li>电动势</li>
<li>闭合电路欧姆定律</li>
</ul>
<h3 id="关于电源电动势和内阻的思考与讨论"><a href="#关于电源电动势和内阻的思考与讨论" class="headerlink" title="关于电源电动势和内阻的思考与讨论"></a>关于电源电动势和内阻的思考与讨论</h3><p><img src="https://cdn.jsdelivr.net/gh/aixuewuli/mdpic@main/img202105/1621862721787-1621862721784.png"><img src="https://cdn.jsdelivr.net/gh/aixuewuli/mdpic@main/img202105/1621865085300-1621865085293.png"></p>
<ul>
<li><p>问题1：如上左图，电压表能测量电源的电动势吗？不能的话，那电压表测的是什么？</p>
</li>
<li><p>问题2：如上右图，能测电源内阻吗？</p>
</li>
</ul>
<details> 
    <summary><font color='blue'>展开查看答案</font></summary> 
    <pre><code>问题1：不能，电压表测的是电路的路端电压，也就是电源的输出电压。如果不能理解那么请确认：1-电压表的示数是电压表的电压（降）；2-电源电动势的含义及与电压的区别 </code></pre> 
    <pre><code>问题2 ：不能，欧姆表的两表笔之间不能接含电源的电路 </code></pre> 
</details>

<p>由此我们得出一个悲观的结论：电源的电动势和内阻不能像我们想当然认为的那样直接测量！</p>
<h3 id="测量电源电动势和内阻的思路"><a href="#测量电源电动势和内阻的思路" class="headerlink" title="测量电源电动势和内阻的思路"></a>测量电源电动势和内阻的思路</h3><p>不能直接测量出E、r ，那么怎么测量？我们先来看看谍战片里常见的一个桥段。</p>
<blockquote>
<p>一黑屋里隐藏着一伙敌特，为制定周密的作战计划，我方需要获取黑屋里敌特人数情报，但又不能打草惊蛇。那么怎么办呢?</p>
</blockquote>
<p>是的，不能进屋直接点人头，但我们可以通过敌特外出采购食物的数量等数据得出屋内人数。一个系统只要与系统外部有相互作用关系，我们能通过内外部的作用关系（需明确具体函数关系，如上例中需明确食物数量与特务人数的函数关系）就可由其外部表现得出其内在。</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">st=&gt;start: 屋外（采购的食物） </span><br><span class="line">e=&gt;end: 屋内（人数）</span><br><span class="line">op1=&gt;operation: 屋内外作用关系（食物与人数的函数关系）</span><br><span class="line">st(right)-&gt;op1(right)-&gt;e</span><br></pre></td></tr></table></figure>

<p>具体步骤：</p>
<ol>
<li>观测特务购买食物的数量</li>
<li>确定食物数量与特务人数间的函数关系</li>
<li>将观测到的购买食物数量代入食物数量与特务人数的函数关系式，求出特务人数。</li>
</ol>
<p>现在我们扯回到测量E、r 上来。我们把待测电源接入到电路中，让电源与外部电路发生作用关系，我们便可根据这个作用关系（具体函数关系是闭合电路欧姆定律），由外部（外部电路的总电压<em>U</em>、总电流 <em>I</em> 及总电阻<em>R</em> 中的两个量）得出其内在（E、r）。</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">st=&gt;start: 外部电路（U、I、R） </span><br><span class="line">e=&gt;end: 电源（E、r）</span><br><span class="line">op1=&gt;operation: 作用关系（闭合电路欧姆定律）</span><br><span class="line">st(right)-&gt;op1(right)-&gt;e</span><br><span class="line"></span><br></pre></td></tr></table></figure>

<h3 id="测量电源电动势和内阻的步骤"><a href="#测量电源电动势和内阻的步骤" class="headerlink" title="测量电源电动势和内阻的步骤"></a>测量电源电动势和内阻的步骤</h3><ol>
<li><p>将电源接入电路（如下图之一）</p>
<p><img src="https://cdn.jsdelivr.net/gh/aixuewuli/mdpic@main/img202105/1621937262446-1621937262439.png"></p>
</li>
<li><p>多次测量外部电路U、I、R中的两个量（需且只需测两个量），相当于观测特务购买食物的数量。图左1测U、I（伏安法） ，图左2测 U、R（伏阻法） ，图左3测I、R（安阻法）。</p>
</li>
<li><p>写出闭合电路欧姆定律，即内外电路各物理量间函数关系，相当于确定食物数量与特务人数间的函数关系。具体由步骤2测量出的两个量加上E、r，这四个量写出闭合电路欧姆定律（ $ E=U+Ir $ 或 $E=U+\frac{U}{R}*r$ 或 $E=IR+Ir$）</p>
</li>
<li><p>数据处理，即根据闭合电路欧姆定律，由测出的外电路物理量求出电源内部E、r。相当于将观测到的购买食物数量代入食物数量与特务人数的函数关系式，求出特务人数。</p>
<p>数据处理方法有两种：</p>
<ul>
<li>方法1联立方程求解：将步骤2中测得的其中两组数据代入到步骤3写出的式子，得到两个方程，联立求出E、r。为减小误差需重复求出几组E、r，然后求平均值。</li>
<li>方法2是图像法求解：利用测量的数据描点绘制出对应图像，如下图所示。</li>
</ul>
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stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 88.5 0 L 88.5 160" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 126.5 0 L 126.5 160" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 138.5 L 160 138.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 63.5 L 160 63.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 25.5 L 160 25.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/></g></g><g id="axis-fb7693ae"><g id="yaxis-fb7693ae"><title>1/U</title><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="fill stroke markers" d=" M 14 3 L 14 160" stroke-opacity="0.9" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray=""/><g class="dcg-svg-axis-line"><path fill="rgb(0,0,0)" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="stroke fill markers" d=" M 14 1.5 L 17.10582854123025 13.09110991546882 L 10.894171458769753 13.09110991546882 L 14 1.5 L 17.10582854123025 13.09110991546882" stroke-opacity="0.9" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray="" fill-opacity="0.9"/></g></g><g id="xaxis-fb7693ae"><title>1/R</title><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="fill stroke markers" d=" M 0 101 L 157 101" stroke-opacity="0.9" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray=""/><g class="dcg-svg-axis-line"><path fill="rgb(0,0,0)" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="stroke fill markers" d=" M 158.5 101 L 146.90889008453118 104.10582854123025 L 146.90889008453118 97.89417145876975 L 158.5 101 L 146.90889008453118 104.10582854123025" stroke-opacity="0.9" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray="" fill-opacity="0.9"/></g></g><g/><g class="dcg-svg-axis-label"><text fill="none" stroke="#ffffff" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="126.326171875" y="112.3046110382404" text-anchor="start" stroke-miterlimit="2" stroke-width="3" stroke-dasharray="">1/R</text><text fill="#000000" stroke="none" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="126.326171875" y="112.3046110382404" text-anchor="start">1/R</text></g><g class="dcg-svg-axis-label" transform="translate(7.704091413580876,33.673828125) rotate(-90,0,0)"><text fill="none" stroke="#ffffff" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="0" y="0" text-anchor="start" stroke-miterlimit="2" stroke-width="3" stroke-dasharray="">1/U</text><text fill="#000000" stroke="none" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="0" y="0" text-anchor="start">1/U</text></g></g></g><g id="expressions-fb7693ae"><g id="sketch-fb7693ae"><title>Expression 1</title><path fill="#2d70b3" stroke="none" paint-order="stroke fill markers" d="" fill-opacity="0.4"/><g><path fill="none" stroke="#2d70b3" class="dcg-svg-curve" paint-order="fill stroke markers" d=" M 13.204091413580876 94.78108162647568 L 13.204091413580876 94.78108162647568 L 156.46277945140326 -5.5" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="2.5" stroke-opacity="0.9" stroke-dasharray=""/></g></g></g><g id="labels-fb7693ae"/></g></svg><br><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="320" height="320"><defs/><g transform="scale(2,2)"><g id="background-88fe8ff7"><rect fill="white" stroke="none" x="0" y="0" width="160" height="160" class="dcg-svg-background"/></g><g id="background-images-88fe8ff7"/><g id="graphpaper-88fe8ff7"><g id="grid-88fe8ff7"><g><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 50.5 0 L 50.5 160" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 88.5 0 L 88.5 160" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 126.5 0 L 126.5 160" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 138.5 L 160 138.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 63.5 L 160 63.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-minor-gridline" paint-order="fill stroke markers" d=" M 0 25.5 L 160 25.5" stroke-opacity="0.2" stroke-miterlimit="10" stroke-dasharray=""/></g></g><g id="axis-88fe8ff7"><g id="yaxis-88fe8ff7"><title>1/ I</title><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="fill stroke markers" d=" M 14 3 L 14 160" stroke-opacity="0.9" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray=""/><g class="dcg-svg-axis-line"><path fill="rgb(0,0,0)" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="stroke fill markers" d=" M 14 1.5 L 17.10582854123025 13.09110991546882 L 10.894171458769753 13.09110991546882 L 14 1.5 L 17.10582854123025 13.09110991546882" stroke-opacity="0.9" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray="" fill-opacity="0.9"/></g></g><g id="xaxis-88fe8ff7"><title>R</title><path fill="none" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="fill stroke markers" d=" M 0 101 L 157 101" stroke-opacity="0.9" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray=""/><g class="dcg-svg-axis-line"><path fill="rgb(0,0,0)" stroke="rgb(0,0,0)" class="dcg-svg-axis-line" paint-order="stroke fill markers" d=" M 158.5 101 L 146.90889008453118 104.10582854123025 L 146.90889008453118 97.89417145876975 L 158.5 101 L 146.90889008453118 104.10582854123025" stroke-opacity="0.9" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="1.5" stroke-dasharray="" fill-opacity="0.9"/></g></g><g/><g class="dcg-svg-axis-label"><text fill="none" stroke="#ffffff" font-family="Times" font-size="14px" font-style="italic" font-weight="normal" text-decoration="normal" x="136.4482421875" y="112.3046110382404" text-anchor="start" stroke-miterlimit="2" stroke-width="3" stroke-dasharray="">R</text><text fill="#000000" stroke="none" font-family="Times" font-size="14px" font-style="italic" font-weight="normal" text-decoration="normal" x="136.4482421875" y="112.3046110382404" text-anchor="start">R</text></g><g class="dcg-svg-axis-label" transform="translate(7.704091413580876,31.67578125) rotate(-90,0,0)"><text fill="none" stroke="#ffffff" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="0" y="0" text-anchor="start" stroke-miterlimit="2" stroke-width="3" stroke-dasharray="">1/ I</text><text fill="#000000" stroke="none" font-family="Arial" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="0" y="0" text-anchor="start">1/ I</text></g></g></g><g id="expressions-88fe8ff7"><g id="sketch-88fe8ff7"><title>Expression 1</title><path fill="#2d70b3" stroke="none" paint-order="stroke fill markers" d="" fill-opacity="0.4"/><g><path fill="none" stroke="#2d70b3" class="dcg-svg-curve" paint-order="fill stroke markers" d=" M 13.204091413580876 82.73402280294627 L 13.204091413580876 82.73402280294627 L 101.43811421652714 -5.5" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="2.5" stroke-opacity="0.9" stroke-dasharray=""/></g></g></g><g id="labels-88fe8ff7"/></g></svg></p>
<p>将步骤3写出的式子变换成一次函数的表达式：$U=-rI+E $  或  $ \frac{1}{U}= \frac{r}{E}*\frac{1}{R}+\frac{1}{E}$  或  $\frac{1}{I}=\frac{1}{E}*R+\frac{r}{E} $（自变量、应变量各自包含步骤2测量的两个量中的一个）。由图像的斜率与截距列方程求出E、r。具体操作看后面的练习题。 </p>
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